A New Algorithm for Multicommodity Flow

TL;DR

This paper introduces a novel algorithm for multicommodity max flow problems that leverages the max-flow min-cut theorem and labeling algorithms, addressing the challenge of extending such theorems to multicommodity networks.

Contribution

The paper presents a new algorithm that adapts single-commodity max flow techniques to multicommodity networks, providing a potential solution to the longstanding max-flow min-cut theorem extension.

Findings

Algorithm successfully computes multicommodity max flow.

Identifies and addresses the difficulty in extending max-flow min-cut theorem.

Provides a systematic approach to record cuts and flows for multiple source-sink pairs.

Abstract

We propose a new algorithm to obtain max flow for the multicommodity flow. This algorithm utilizes the max-flow min-cut theorem and the well known labeling algorithm due to Ford and Fulkerson [1]. We proceed as follows: We select one source/sink pair among the n distinguished source/sink pairs at a time and treat the given multicommodity network as a single commodity network for such chosen source/sink pair. Then applying standard labeling algorithm, separately for each sink/source pair, the feasible flow which is max flow and the corresponding minimum cut corresponding to each source/sink pair is obtained. A record is made of these cuts and the paths flowing through the edges of these cuts. This record is then utilized to develop our algorithm to obtain max flow for multicommodity flow. In this paper we have pinpointed the difficulty behind not getting a max flow min cut type theorem for multicommodity flow and found out a remedy.